Purely infinite simple Leavitt path algebras

By G. Abrams, G. Pino. 2005

87 citations recorded by the source corpus.

Abstract We give necessary and sufficient conditions on a row-finite graph E so that the Leavitt path algebra L ( E ) is purely infinite simple. This result provides the algebraic analog to the corresponding result for the Cuntz–Krieger C ∗ -algebra C ∗ ( E ) given in [T. Bates, D. Pask, I. Raeburn, W. Szymanski, The C ∗ -algebras of row-finite graphs, New York J. Math. 6 (2000) 307–324].

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/146f13e45df18aacb32345f83de2be6119fae0d5

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Citations in the corpus

The graded Grothendieck group and the classification of Leavitt path algebras (2011, background): “THE GRADED GROTHENDIECK GROUP AND LEAVITT PATH ALGEBRAS 17 This theorem shows that many interesting classes of Leavitt path algebras fall into the category of strongly graded LPAs, such as purely infinite simple Leavitt path algebras (see [2]).”
Isomorphism and Morita equivalence of graph algebras (2008, background): “We refer the reader to [13] and [5] for the appropriate definitions of the above terms as well as proofs that any E in Fn has the property that the graph algebras C∗(E) and LC(E) are both purely infinite and simple.”
Exchange Leavitt path algebras and stable rank (2005, methodology): “(2) A graph E satisfies condition (K) if for each vertex v on a closed simple path there exists at least two distinct closed simple paths α, β based at v, or, following [ 2 ], V1 = ∅.”
Locally finite Leavitt path algebras (2008, unknown): “(2) r(e)e∗ = e∗s(e) = e∗ for all e ∈ E(1).”
Finite-dimensional Leavitt path algebras (2007, background): “In [1], [2] and [3], a somewhat different description of LK(E) is given (in terms of quotients of path algebras).”
Uniqueness theorems and ideal structure for Leavitt path algebras (2006, background): “Abrams and Aranda-Pino established basic properties of these Leavitt path algebras, and soon after there was a flurry of activity as many other authors [2, 3, 4, 5, 6, 7, 8] investigated their structure and found applications to various topics in algebra.”
Chain conditions for Leavitt path algebras (2009, background): “For example, conditions on the graph allow us to decide when the corresponding Leavitt path algebra is simple [1], purely infinite simple [2], exchange [9], finite-dimensional [4].”
Socle theory for Leavitt path algebras of arbitrary graphs (2008, background): “In both situations the classification of simple ([1]) and purely infinite simple ([ 2 ]) structures was carried out in terms of properties of the graph.”
Leavitt path algebras: the first decade (2014, methodology): “(Simplicity and Purely Infinite Simplicity Theorems for graph C∗algebras) Let E be a finite graph.”